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We present a detailed orbital and stability analysis of the HD~59686binary-star planet system. HD~59686 is a single-lined moderately close ($a_{B}= 13.6\,$AU) eccentric ($e_{B} = 0.73$) binary, where the primary is an evolvedK giant with mass $M = 1.9 M_{\odot}$ and the secondary is a star with aminimum mass of $m_{B} = 0.53 M_{\odot}$. Additionally, on the basis of preciseradial velocity (RV) data a Jovian planet with a minimum mass of $m_p = 7M_{\mathrm{Jup}}$, orbiting the primary on a nearly circular S-type orbit with$e_p = 0.05$ and $a_p = 1.09\,$AU, has recently been announced. We investigatelarge sets of orbital fits consistent with HD 59686's radial velocity data byapplying bootstrap and systematic grid-search techniques coupled withself-consistent dynamical fitting. We perform long-term dynamical integrationsof these fits to constrain the permitted orbital configurations. We find thatif the binary and the planet in this system have prograde and aligned coplanarorbits, there are narrow regions of stable orbital solutions locked in asecular apsidal alignment with the angle between the periapses, $\Delta\omega$, librating about $0^\circ$. We also test a large number of mutuallyinclined dynamical models in an attempt to constrain the three-dimensionalorbital architecture. We find that for nearly coplanar and retrograde orbitswith mutual inclination $145^\circ \lesssim \Delta i \leq 180^\circ$, thesystem is fully stable for a large range of orbital solutions.

Dynamical analysis of the circumprimary planet in the eccentric binary system HD59686